An Attempt to Derive the epsilon Equation from a Two-Point Closure

The goal of this paper is to derive the equation for the turbulence dissipation rate epsilon for a shear-driven flow. In 1961, Davydov used a one-point closure model to derive the epsilon equation from first principles but the final result contained undetermined terms and thus lacked predictive power. Both in 1987 and in 2001, attempts were made to derive the epsilon equation from first principles using a two-point closure, but their methods relied on a phenomenological assumption. The standard practice has thus been to employ a heuristic form of the equation that contains three empirical ingredients: two constants, c(sub 1 epsilon), and c(sub 2 epsilon), and a diffusion term D(sub epsilon) In this work, a two-point closure is employed, yielding the following results: 1) the empirical constants get replaced by c(sub 1), c(sub 2), which are now functions of Kappa and epsilon; 2) c(sub 1) and c(sub 2) are not independent because a general relation between the two that are valid for any Kappa and epsilon are derived; 3) c(sub 1), c(sub 2) become constant with values close to the empirical values c(sub 1 epsilon), c(sub epsilon 2), (i.e., homogenous flows); and 4) the empirical form of the diffusion term D(sub epsilon) is no longer needed because it gets substituted by the Kappa-epsilon dependence of c(sub 1), c(sub 2), which plays the role of the diffusion, together with the diffusion of the turbulent kinetic energy D(sub Kappa), which now enters the new equation (i.e., inhomogeneous flows). Thus, the three empirical ingredients c(sub 1 epsilon), c(sub epsilon 2), D (sub epsilon)are replaced by a single function c(sub 1)(Kappa, epsilon ) or c(sub 2)(Kappa, epsilon ), plus a D(sub Kappa)term. Three tests of the new equation for epsilon are presented: one concerning channel flow and two concerning the shear-driven planetary boundary layer (PBL).